There are 4 different types of damage you can be dealt by an enemy:
Kinetic: Just hammering into you at speed, like a bullet.
Thermal: Melting you, like a flamethrower.
Explosive: Blowing up on impact, like a missile.
Electromagnetic (EM): Other types of radiation, like microwave and gamma radiation, like an atomic blast.
Logically enough most types of weapon deal more than one type of damage at once, eg:
Hybrid weapons deal kinetic and thermal damage.
Projectile weapons can deal explosive and thermal damage.
Laser weapons deal EM and thermal damage.
... and specific weapons and charges will deal different balances of the two types. The exception to this is missiles, which are specifically designed to deal one type of damage, and one only, though that can be any of the four types. Missile ships do however usually have a bonus to kinetic damage (since they are being thrown at the enemy anyway) so NPC missile throwers pretty much universally use kinetic missiles.
This means most enemies are fairly predictable, e.g.
Guristas use Caldari ships, firing hybrid weapons and missiles, damage will be Kin/Therm.
Angels use Minmatar ships, firing projectile weapons, damage will be Exp/Therm.
Sanshas use Amarr ships, firing lasers, damage will be EM/Therm.
Serpentis use Gallente ships firing hybrid weapons, damage will be Therm/Kin.
All very fascinating, you may be thinking, but what does this have to do with me? Most ships have bonuses to one type of weapon, so you are stuck with the one type of damage. This can be gotten around but where it affects you most is in your ships ability to deal with that damage.
To take shields as an example: Typically shields have the weakest protection against EM radiation, because it interferes with the electronics projecting them. If you go to your fitting screens you will see that by default most shields have 0% resistance against EM radiation, which means that an EM weapon dealing 100 damage will reduce your hitpoints by 100. Equally you will see, however, that an explosive weapon hitting the shield only does 50% of its damage. This means that against explosive damage, your shields are effectively twice as effective! To get the same effect for EM we would have to double the size of our shields. Or would we?
Actually there are specific mods that will increase our resistance to damage much more easily than simply increasing the size of our shields:
Shield resistance amplifiers are passive modules (i.e. you don't have to turn them on, and they don't use any capacitor) that give a huge boost to resistance. They only cover one type of resistance each and have a large penalty for using a second of the same type. They also have very low fitting requirements. Best used on small ships like frigates and destroyers.
Shield Hardeners are active modules that typically give an equally large boost to resistance when switched on, and a slightly lesser one when inactive. They have much less of a penalty for using more than one and some will protect against all types of damage. Used on all larger ships where the capacitor cost is insignificant.
You can also boost the effectiveness of these modules, particularly the passive ones, using skills; look at the shield compensation skills in the market.
The calculation for the effectiveness of these resistance mods is quite complex but the easiest way to think of it is that these mods increase your resistance by the percentage shown in their info but only taking your remaining unprotected percentage into account. For instance a 50% EM resistance mod will increase resistance on EM to 50%. The same kind of mod for explosive however will only increase resistance to around 75%. It has increased resistance by 50% of the remaining 50%. A second (active) hardener will increase it to around 87%, and so on. You can easily see therefeore that resistance never gets up to 100%; the most you can hope for is around 92%. This would mean however that an enemy that previously hit for 100 would be hitting for 8, which is much less threatening. When combined with boosters that increase a fixed amount of shield this will drastically increase the amount of damage you can withstand. Shields also have a natural recharge which will help more the greater your resistance.
So much for shields. For armor tankers the procedure is much the same;
Resistance or Energised platings are the passive modules (resistance require fewer skills but are less effective). Armor tankers, unlike shield tankers, have adaptive modules that can protect against all damage types, to a greater or lesser extent, but still require no capacitor usage.
Armor hardeners are the active modules.
For armor the relevant skills are armor compensation.
Finally there is structure. Typically structure has zero resistance to all damage types meaning by the time you are into structure it dissapears in a flash. Experienced pilots keep a Damage Control as part of their setup: this will not only at least double your structure resistances and therefore your survival time should the worst happen but also give a small bonus to armor and shield resists, which is unaffected by the stacking penalty. Damage controls are an active mod, but the cost is insignificant.
Using resistances to increase your damage is MUCH more effective than merely increasing the size of your shields/armor. A 90% resistance ship has effectively 10x the hitpoints of a ship without. Try increasing your shield by a factor of 10 without the use of resistances. Increasing the size of your shields also encreases the size of your ship, making you easier to hit. For armor increasing the size of your armor plating will make your ship heavier, and slow it down, also making it easier to hit (and also making it painful to get anywhere).